Several experimental and theoretical studies report instances of concerted or correlated multiple proton tunneling in solid phases of water. Here, we construct a pseudo-spin model for the quantum motion of protons in a hexameric H$_2$O ring and extend it to open system dynamics that takes environmental effects into account in the form of O$-$H stretch vibrations. We approach the problem of correlations in tunneling using quantum information theory in a departure from previous studies. Our formalism enables us to quantify the coherent proton mobility around the hexagonal ring by one of the principal measures of coherence, the $l_1$ norm of coherence. The nature of the pairwise pseudo-spin correlations underlying the overall mobility is further investigated within this formalism. We show that the classical correlations of the individual quantum tunneling events in long-time limit is sufficient to capture the behaviour of coherent proton mobility observed in low-temperature experiments. We conclude that long-range intra-ring interactions do not appear to be a necessary condition for correlated proton tunneling in water ice.
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